Molar volume of condensed phase

Because, at constant temperature, dGm = Vm dP and the molar volumes of condensed phases are very small, it is usually sufficiently accurate to take their molar free energy as pressure independent and the same as that at the 1.0-bar standard state. This is equivalent to setting the activity of pure, condensed phases equal to unity. (See Problem 9.) The activity of a condensed phase is also independent of just how much of the phase is present. As a result of these considerations, no variable describing the condensed phase appears in the equilibrium constant and the equilibrium is independent of just how much condensed phase is present. 

Strictly speaking the partial pressures depend on the total pressure, but it is readily shown, as in 1, that the variation is negligible at ordinary pressure because of the small molar volumes of condensed phases. Equation (21.53) therefore reduces to the Duhem-Margules equation, ... 

It is also useful to note that several simplifications can be made in computing the thermodynamic propenies of solids and liquids. First, because the molar volumes of condensed phases are small, the product PV can be neglected unless the pressure is high. Thus, for solids and liquids. 

Molar volumes of condensed phases are much smaller than those of gases, and to a good approximation we may write... 

According to quantum mechanics, isolated molecules do not have a finite boundary, but rather fade away into the regions of low electron density. It has been well established, however, from properties of condensed matter and molecular interactions, that individual molecules occupy a finite and measurable volume. This notion is at the core of the concept of molecular structure. 33 A number of physical methods yield estimations of molecular dimensions. These methods include measurements of molar volumes in condensed phases, critical parameters (lattice spacings and bond distances), and collision diameters in the gas phase. 34 From these results, one derives values of atomic radii from which a number of empirical molecular surfaces can be built. Note that the values of the atomic radii depend on the physical measurement chosen. 35-i37... 

The molar volumes of CH4 and CD4 are known to differ in the expected direction in the condensed phases at low temperatures (Olusius and Weigand, 1940 Grigor and Steele, 1968). Such a difference would in principle originate in the movement of the molecule as a whole as well as in its carbon-hydrogen oscillations, but a macroscopically available compound would be expected to offer better possibilities for experimental investigation than poorly known transition states. 

The effect of curvature is much more pronounced for the thermodynamics of a gas bubble than for the liquid droplet. The curvature is a pressure effect, which is much larger for gases than for condensed phases, reflecting the much larger molar volume of the gas. 

Compressing ammonia gas under high pressure forces the molecules into close proximity. In a normal gas, the separation between each molecule is generally large - approximately 1000 molecular diameters is a good generalization. By contrast, the separation between the molecules in a condensed phase (solid or liquid) is more likely to be one to two molecular diameters, thereby explaining why the molar volume of a solid or liquid is so much smaller than the molar volume of a gas. 

Abstract Isotope effects on the PVT properties of non-ideal gases and isotope effects on condensed phase physical properties such as vapor pressure, molar volume, heats of vaporization or solution, solubility, etc., are treated in some thermodynamic detail. Both pure component and mixture properties are considered. Numerous examples of condensed phase isotope effects are employed to illustrate theoretical and practical points of interest. 

Although Equations (8.28) and (8.32) are formally alike, they refer to different types of processes. The former is strictly true for a process that occurs at a constant pressure throughout a temperature range. Vaporization or sublimation does not fulfill this restriction, but nevertheless. Equation (8.32) is approximately correct because the molar volume of the condensed phase is small compared with that of the gas, and the vapor pressure is small enough that the vapor behaves as an ideal gas. 

Table II presents some consequences of these analogies for a vapor phase and a condensed phase together with the thermal analogs. Here rti is the concentration of the vapor in molecules per unit volume, Nt is the mole fraction in the condensed phase, Vc is the molar volume of the substrate, p< is the partial pressure, and y< is the rational activity coefficient.

The Clapeyron equation can be simplified to some extent for the case in which a condensed phase (liquid or solid) is in equilibrium with a gas phase. At temperatures removed from the critical temperature, the molar volume of the gas phase is very much larger than the molar volume of the condensed phase. In such cases the molar volume of the condensed phase may be neglected. An equation of state is then used to express the molar volume of the gas as a function of the temperature and pressure. When the virial equation of state (accurate to the second virial coefficient) is used,... 

If the partial molar volumes of the condensed phase are negligibly small with respect to the molar volume of the gas phase, this becomes... 

We see that under these conditions the value of the derivative becomes zero (a maximum) when xx = 2x2. The maximum then occurs when the composition of the condensed phase is that of the species A2B. When the partial molar volumes of the condensed phase are not negligibly small and when the nonideality of the gas is included, the maximum does not occur at this exact composition, but does occur very close to it [32]. 

In Fig. 2, the chemical potential curves of Fig. la are shown for two different pressures. Because the molar volume of a gas is greater than that of condensed phases, the chemical potential of the gas is increased much more than those of liquid or solid by increasing pressure. The boiling point and sublimation point therefore increase with pressure. The molar volume of the liquid and solid are comparable, and either one may be larger. As a result, the melting point may either increase or decrease with pressure. 

The chemical potentials of condensed phases are relatively pressure independent, due to their small molar volumes.) Rewriting,... 

Since a Maxwell relation for equation (6) shows that the partial molar volume of the condensed phase is... 

The isotherm (plot of pressure versus volume) at the temperature Tc plays a special role in the theory of the states of matter. An isotherm behaves in accordance with the gas laws slightly below T. At certain pressure, a liquid condenses from gaseous state and is distinguishable from it by the presence of a visible interface. If, however, the compression takes place at a surface separating two phases does not appear and the volumes at each end of the horizontal part of the isotherm have merged to a single point, critical point of the gas. The temperature, pressure, and molar volume at the critical point are called the critical temperature T, critical pressure P, and critical molar volume of the substance respectively. Collectively,... 

This equation can be put in a simpler approximate form if it is assumed that the vapour phase behaves as a perfect gas, and if the molar volume of the condensed phase is neglected in comparison with that of the vapour. With those assumptions... 

Jacob et al. used the method of characteristics to discuss the general properties of the system of mass balance equations in multicomponent preparative gas chromatography (GC) [34-36], assuming either a linear or a nonlinear isotherm. The GC problem is more complicated than the HPLC one because the gas mobile phase is much more compressible than a solution and the mobile phase velocity is very different inside and outside a high concentration band because the partial molar volumes of compounds are much larger in the gas mobile phase than in the condensed stationary phase (the sorption effect). They showed that the method of characteristics appHes to multicomponent systems as well as to single component... 

Since the molar volume of a condensed phase is frequently insensitive lo pressure. Eq. (L.2-31) cen often he approximated by... 

In order to see the dependence of AG on the dimensions of the condensed particle, let us substitute for n the number of moles of condensed vapor, n = 4irr /3V, where V = M/p is the molar volume of the condensed phase, M is its atomic weight, and p is the density. We have... 

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